Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C23×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,15)(11,13)(12,14)(16,17,19,24,21,25,20,23)(18,22,27,30,28,29,26,31);(8,10,12,11)(9,13,14,15)(16,18)(17,22)(19,27)(20,26)(21,28)(23,31)(24,30)(25,29);(8,11,12,10)(9,13,14,15)(17,23)(18,26)(19,20)(22,29)(24,25)(27,28)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5324);